Journal of Inequalities in Pure and
Applied Mathematics
MAXIMAL OPERATORS OF FEJÉR MEANS OF
VILENKIN-FOURIER SERIES
volume 7, issue 4, article 149,
2006.
ISTVÁN BLAHOTA, GYÖRGY GÁT AND USHANGI GOGINAVA
Institute of Mathematics and Computer Science
College of Nyíregyháza
P.O. Box 166, Nyíregyháza
H-4400 Hungary
EMail: blahota@nyf.hu
EMail: gatgy@nyf.hu
Department of Mechanics and Mathematics
Tbilisi State University
Chavchavadze str. 1
Tbilisi 0128, Georgia
EMail: z_goginava@hotmail.com
Received 12 June, 2006;
accepted 22 November, 2006.
Communicated by: Zs. Páles
Abstract
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2000
Victoria University
ISSN (electronic): 1443-5756
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Abstract
The main aim of this paper is to prove that the maximal operator σ∗ := sup |σn |
n
of the Fejér means of the Vilenkin-Fourier series is not bounded from the Hardy
space H1/2 to the space L1/2 .
2000 Mathematics Subject Classification: 42C10.
Key words: Vilenkin system, Hardy space, Maximal operator.
The first author is supported by the Békésy Postdoctoral fellowship of the Hungarian
Ministry of Education Bö 91/2003, the second author is supported by the Hungarian
National Foundation for Scientific Research (OTKA), grant no. M 36511/2001., T
048780 and by the Széchenyi fellowship of the Hungarian Ministry of Education Szö
184/2003.
Maximal Operators of Fejér
Means of Vilenkin-Fourier
Series
István Blahota, György Gát and
Ushangi Goginava
Title Page
Contents
Let N+ denote the set of positive integers, N := N+ ∪ {0}. Let m :=
(m0 , m1 , . . . ) denote a sequence of positive integers not less than 2. Denote
by Zmk := {0, 1, . . . , mk − 1} the additive group of integers modulo mk .
Define the group Gm as the complete direct product of the groups Zmj , with
the product of the discrete topologies of Zmj ’s.
The direct product µ of the measures
µk ({j}) :=
1
mk
(j ∈ Zmk )
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J. Ineq. Pure and Appl. Math. 7(4) Art. 149, 2006
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is the Haar measure on Gm with µ(Gm ) = 1.
If the sequence m is bounded, then Gm is called a bounded Vilenkin group,
else it is called an unbounded one. The elements of Gm can be represented by
sequences x := (x0 , x1 , . . . , xj , . . . ) (xj ∈ Zmj ). It is easy to give a base for the
neighborhoods of Gm :
I0 (x) := Gm ,
In (x) := {y ∈ Gm |y0 = x0 , . . . , yn−1 = xn−1 }
for x ∈ Gm , n ∈ N. Define In := In (0) for n ∈ N+ .
If we define the so-called generalized number system based on m in the
following way:
Mk+1 := mk Mk (k ∈ N),
P
then every n ∈ N can be uniquely expressed as n = ∞
j=0 nj Mj , where nj ∈
Zmj (j ∈ N+ ) and only a finite number of nj ’s differ from zero. We use the
following notations. Let (forPn > 0) |n| := max{k ∈ N : nk 6= 0} (that is,
(k)
M|n| ≤ n < M|n|+1 ), n(k) = ∞
j=k nj Mj and n(k) := n − n .
Denote by Lp (Gm ) the usual (one dimensional) Lebesgue spaces (k · kp the
corresponding norms) (1 ≤ p ≤ ∞).
Next, we introduce on Gm an orthonormal system which is called the Vilenkin
system. At first define the complex valued functions rk (x) : Gm → C, the generalized Rademacher functions as
Maximal Operators of Fejér
Means of Vilenkin-Fourier
Series
István Blahota, György Gát and
Ushangi Goginava
M0 := 1,
2πıxk
rk (x) := exp
mk
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2
(ı = −1, x ∈ Gm , k ∈ N).
J. Ineq. Pure and Appl. Math. 7(4) Art. 149, 2006
http://jipam.vu.edu.au
Now define the Vilenkin system ψ := (ψn : n ∈ N) on Gm as:
ψn (x) :=
∞
Y
rknk (x)
(n ∈ N).
k=0
Specifically, we call this system the Walsh-Paley one if m ≡ 2.
The Vilenkin system is orthonormal and complete in L1 (Gm ) [9].
Now, we introduce analogues of the usual definitions in Fourier-analysis. If
f ∈ L1 (Gm ) we can establish the following definitions in the usual manner:
Z
b
(Fourier coefficients)
f (k) :=
f ψ k dµ
(k ∈ N),
Gm
(Partial sums)
Sn f :=
n−1
X
(n ∈ N+ , S0 f := 0),
fb(k)ψk
k=0
n−1
(Fejér means)
σn f :=
1X
Sn f
n k=0
(n ∈ N+ ),
Maximal Operators of Fejér
Means of Vilenkin-Fourier
Series
István Blahota, György Gát and
Ushangi Goginava
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(Dirichlet kernels)
Dn :=
n−1
X
k=0
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ψk
(n ∈ N+ ).
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Recall that
(
DMn (x) =
(1)
Mn , if x ∈ In ,
0,
if x ∈ Gm \In .
The norm (or quasinorm) of the space Lp (Gm ) is defined by
Z
kf kp :=
p1
|f (x)| µ (x)
p
(0 < p < +∞) .
Gm
The space weak-Lp (Gm ) consists of all measurable functions f for which
1
kf kweak−Lp (Gm ) := sup λµ (|f | > λ) p < +∞.
Maximal Operators of Fejér
Means of Vilenkin-Fourier
Series
István Blahota, György Gát and
Ushangi Goginava
λ>0
The σ-algebra generated by the intervals {In (x) : (x) ∈ Gm } will be denoted by Fn (n ∈ N) .
Denote by f = f (n) , n ∈ N a martingale with respect to (Fn , n ∈ N) (for
details see, e. g. [10, 14]).
The maximal function of a martingale f is defined by
f ∗ = sup f (n) ,
n∈N
respectively.
In case f ∈ L1 (Gm ), the maximal functions are also be given by
Z
1
∗
f (x) = sup
f (u) µ (u) .
n∈N µ (In (x))
In (x)
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For 0 < p < ∞ the Hardy martingale spaces Hp (Gm ) consist of all martingales for which
kf kHp := kf ∗ kp < ∞.
If f ∈ L1 (Gm ) , then it is easy to show that the sequence (SMn (f ) : n ∈ N)
is a martingale. If f is a martingale, that is f = (f (n) : n ∈ N), then the
Vilenkin-Fourier coefficients must be defined in a slightly different manner:
Z
b
f (i) = lim
f (k) (x) ψ i (x) µ (x) .
k→∞
Gm
The Vilenkin-Fourier coefficients of f ∈ L1 (Gm ) are the same as those of
the martingale (SMn (f ) : n ∈ N) obtained from f .
For a martingale f the maximal operators of the Fejér means are defined by
σ ∗ f (x) = sup |σn (f ; x)|.
n∈N
In this one-dimensional case the weak type inequality
µ (σ ∗ f > λ) ≤
c
kf k1
λ
(λ > 0)
can be found in Zygmund [16] for the trigonometric series, in Schipp [6] for
Walsh series and in Pál, Simon [5] for bounded Vilenkin series. Again in onedimension, Fujji [3] and Simon [8] verified that σ ∗ is bounded from H1 to L1 .
Weisz [11, 13] generalized this result and proved the boundedness of σ ∗ from
the martingale Hardy space Hp to the space Lp for p > 1/2. Simon [7] gave a
counterexample, which shows that this boundedness does not hold for 0 < p <
1/2. In the endpoint case p = 1/2 Weisz [15] proved that σ ∗ is bounded from
Maximal Operators of Fejér
Means of Vilenkin-Fourier
Series
István Blahota, György Gát and
Ushangi Goginava
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the Hardy space H1/2 to the space weak-L1/2 . By interpolation it follows that
σ ∗ is not bounded from Hp to the space weak-Lp for all 0 < p < 1/2.
Theorem 1. For any bounded Vilenkin system the maximal operator σ ∗ of the
Fejér means is not bounded from the Hardy space H1/2 to the space L1/2 .
The Fejér kernel of order n of the Vilenkin-Fourier series is defined by
n−1
Kn (x) :=
1X
Dk (x) .
n k=0
In order to prove the theorem we need the following lemmas.
Lemma 2 ([4]). Suppose that s, t, n ∈ N and x ∈ It \It+1 . If t ≤ s ≤ |n|, then
(n(s+1) + Ms )Kn(s+1) +Ms (x) − n(s+1) Kn(s+1)
(
Mt Ms ψn(s+1) (x) 1−r1t (x) ,
=
0,
Maximal Operators of Fejér
Means of Vilenkin-Fourier
Series
István Blahota, György Gát and
Ushangi Goginava
Title Page
if x − xt et ∈ Is ,
otherwise.
n∗A
Lemma 3 ([2]). Let 2 < A ∈ N+ , k ≤ s < A and
:= M2A + M2A−2 + · · · +
M2 + M0 . Then
M M
∗
2k
2s
∗
nA−1 KnA−1 (x) ≥
4
for
x ∈ I2A (0, . . . , 0, x2k 6= 0, 0, . . . , 0, x2s 6= 0, x2s+1 , . . . , x2A−1 ),
k = 0, 1, . . . , A − 3,
s = k + 2, k + 3, . . . , A − 1.
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Proof of Theorem 1. Let A ∈ N+ and
fA (x) := DM2A+1 (x) − DM2A (x) .
In the sequel we are going to prove for the function fA that
kσ ∗ fA k1/2
≥ c log2q A,
kfA kH1/2
where q = sup {m0 , m1 , . . . } and constant c depends only on q. This inequality
obviously would show the unboundedness of σ ∗ .
It is evident that
(
1, if i = M2A , . . . , M2A+1 − 1,
fbA (i) =
0, otherwise.
Then we can write


Di (x) − DM2A (x) , if i = M2A + 1, . . . , M2A+1 − 1,



fA (x) ,
if i ≥ M2A+1 ,
(2) Si (fA ; x) =



 0,
otherwise.
Since
Maximal Operators of Fejér
Means of Vilenkin-Fourier
Series
István Blahota, György Gát and
Ushangi Goginava
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fA∗ (x) = sup |SMn (fA ; x)| = |fA (x)| ,
n∈N
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from (1) we get
(3)
kfA kH1/2 = kfA∗ k1/2 = DM2A+1 − DM2A 1/2
!2
Z
Z
1
1
2
=
M2A
+
|M2A+1 − M2A | 2
I2A \I2A+1
I2A+1
1
(m2A − 1) 2 12
m2A − 1 12
M2A
M2A +
M2A+1
M2A+1
=
≤
≤
!2
−1
2 m2A M2A
−1
cM2A
.
2
Maximal Operators of Fejér
Means of Vilenkin-Fourier
Series
István Blahota, György Gát and
Ushangi Goginava
Since
Dk+M2A − DM2A = ψM2A Dk ,
k = 1, 2, . . . , M2A ,
from (2) we obtain
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(4)
∗
σ fA (x) = sup |σn (fA ; x)|
n∈N
≥ σn∗A (fA ; x)
∗
nX
A −1
1 = ∗ Si (fA ; x)
nA i=0
∗
nX
A −1
1 = ∗ (Di (x) − DM2A (x))
nA i=M +1
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2A
J. Ineq. Pure and Appl. Math. 7(4) Art. 149, 2006
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∗
nA−1 −1
X
1 = ∗ (Di+M2A (x) − DM2A (x))
nA i=1
∗
n
∗
K
(x) .
= A−1
∗ nA−1
nA
h
√ i
Let q := sup{mi : i ∈}. For every l = 1, . . . , 14 logq A −1 (A is supposed
to be large enough) let kl be the smallest natural numbers, for which
√ 1
√
1
2
M2A A 4l ≤ M2k
< M2A A 4l−4
l
q
q
István Blahota, György Gát and
Ushangi Goginava
hold.
Denote
k,s
I2A
(x) := I2A (0, . . . , 0, x2k 6= 0, 0, . . . , 0, x2s 6= 0, x2s+1 , . . . , x2A−1 )
Title Page
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and let
x∈
kl ,kl +1
I2A
(z)
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Then from Lemma 3 and (4) we obtain that
√
2
M2k
A
l
σ fA (x) ≥ c
≥ c 4l
M2A
q
∗
√
kσ ∗ fA k1/2 ≥ c
A] m2kl +3 −1
[ 41 log
q
X
X
l=1
x2kl +3 =0
m2A−1 −1
···
Close
√
4
X
x2A−1 =0
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On the other hand,
q
Maximal Operators of Fejér
Means of Vilenkin-Fourier
Series
q
A kl ,kl +1
µ I2A
(x)
2l
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J. Ineq. Pure and Appl. Math. 7(4) Art. 149, 2006
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√
√
4
≥c A
√
4
=c A
√
4
≥c A
√
4
≥c A
A]
[ 14 log
q
X
1
4
l=1
√
m2kl +3 · · · m2A−1
q 2l M2A
[ X A]
logq
l=1
√
logq A]
[ 14 X
l=1
√
logq A]
1
q 2l M
2kl +2
1
q 2l M2kl
[ 14 X
l=1
q 2l
q
Maximal Operators of Fejér
Means of Vilenkin-Fourier
Series
1
√
M2A
István Blahota, György Gát and
Ushangi Goginava
Aq −4l+4
logq A
.
≥ c√
M2A
Title Page
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Combining this with (3) we obtain
c log2q A
kσ ∗ fA k1/2
≥
M2A = c log2q A → ∞
kfA kH1/2
M2A
Thus, the theorem is proved.
as A → ∞.
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References
[1] G.N. AGAEV, N.Ya. VILENKIN, G.M. DZHAFARLI AND A.I. RUBINSHTEJN, Multiplicative systems of functions and harmonic analysis on
zero-dimensional groups, Baku, Ehlm, 1981 (in Russian).
[2] I. BLAHOTA, G. GÁT AND U. GOGINAVA, Maximal operators of Fejér
means of double Vilenkin-Fourier series, Colloq. Math., to appear.
[3] N.J. FUJII, Cesàro summability of Walsh-Fourier series, Proc. Amer.
Math. Soc., 77 (1979), 111–116.
[4] G. GÁT, Pointwise convergence of the Fejér means of functions on unbounded Vilenkin groups, J. of Approximation Theory, 101(1) (1999), 1–
36.
[5] J. PÁL AND P. SIMON, On a generalization of the concept of derivate,
Acta Math. Hung., 29 (1977), 155–164.
[6] F. SCHIPP, Certain rearrangements of series in the Walsh series, Mat. Zametki, 18 (1975), 193–201.
[7] P. SIMON, Cesaro summability with respect to two-parameter Walsh system, Monatsh. Math., 131 (2000), 321–334.
[8] P. SIMON, Investigations with respect to the Vilenkin system, Annales
Univ. Sci. Budapest Eötv., Sect. Math., 28 (1985), 87–101.
[9] N. Ya. VILENKIN, A class of complete orthonormal systems, Izv. Akad.
Nauk. U.S.S.R., Ser. Mat., 11 (1947), 363–400
Maximal Operators of Fejér
Means of Vilenkin-Fourier
Series
István Blahota, György Gát and
Ushangi Goginava
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[10] F. WEISZ, Martingale Hardy Spaces and their Applications in Fourier
Analysis, Springer, Berlin - Heidelberg - New York, 1994.
[11] F. WEISZ, Cesàro summability of one and two-dimensional Walsh-Fourier
series, Anal. Math., 22 (1996), 229–242.
[12] F. WEISZ, Hardy spaces and Cesàro means of two-dimensional Fourier
series, Bolyai Soc. Math. Studies, 5 (1996), 353–367.
[13] F. WEISZ, Bounded operators on Weak Hardy spaces and applications,
Acta Math. Hungar., 80 (1998), 249–264.
Maximal Operators of Fejér
Means of Vilenkin-Fourier
Series
[14] F. WEISZ, Summability of Multi-dimensional Fourier Series and Hardy
Space, Kluwer Academic, Dordrecht, 2002.
István Blahota, György Gát and
Ushangi Goginava
[15] F. WEISZ, ϑ-summability of Fourier series, Acta Math. Hungar., 103(1-2)
(2004), 139–176.
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[16] A. ZYGMUND, Trigonometric Series, Vol. 1, Cambridge Univ. Press,
1959.
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